Double Bubbles on the Line with Log-Convex Density $f$ with $(\log f)'$ Bounded

We extend results of Bongiovanni et al. on double bubbles on the line with log-convex density to the case where the derivative of the log of the density is bounded. We show that the tie function between the double interval and the triple interval still exists but may blow up to infinity in finite time. For the first time, a density is presented for which the blowup time is positive and finite.